Shear Strain Localization in Plastic Deformations

  • Athanasios E. Tzavaras
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 52)


Shear instabilities in the form of shear bands are often observed during high speed, plastic deformations of metals. According to one theory, their formation is attributed to effective strain-softening response, which results at high strain rates as the net outcome of the influence of thermal softening on the, normally, strain-hardening response of metals. In order to test the core instability mechanism set forth by such theories, we consider one-dimensional shear deformations of a material exhibiting strain softening and strain-rate sensitivity. The deformation is caused by either prescribed tractions or prescribed velocities. As it turns out, for moderate amounts of strain softening, strain-rate sensitivity exerts a dissipative effect and stabilizes the motion. However, once a threshold is exceeded, the response becomes unstable and shear strain localization can occur.


Shear Band High Strain Rate Strain Softening Thermal Softening Maximal Interval 
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  1. BPV] M. Bertsch, L. A. Peletier and S. M. Verduyn Lunel, The effect of temperature dependent viscosity on shear flow of incompressible fluids, SIAM J. Math. Anal, (to appear).Google Scholar
  2. [C]
    N. Charalambakis, Stabilite asymptotique lorsque t → ∞ en thermoviscoelasticité et ther- moplasticité, in “Nonlinear partial differential equations and their applications: Collège de France Seminar Vol IX”, (H. Brezis and J.-L. Lions, eds.), Pitman Res. Notes in Math. No 181, Longman Scientific and Technical, Essex, England, 1988.Google Scholar
  3. [CM]
    N. Charalambakis and F. Murat, Weak solutions to the initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials, Proc. Royal Soc. Edinburgh 113A (1989), 257–265.MathSciNetMATHGoogle Scholar
  4. [CDM]
    H.T. Chen, A.S. Douglas and R. Malek-Madani, An asymptotic stability condition for inhomogeneous for inhomogeneous simple shear, Quart. Appi. Math. 47 (1989), 247–262.MathSciNetMATHGoogle Scholar
  5. [CCS]
    K. Chueh, C. conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J. 26 (1977), 372–411.MathSciNetCrossRefGoogle Scholar
  6. [D1]
    C. M. Dafermos, Global smooth solutions to the initial boundary value problem of one- dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Analysis 13 (1982), 397–408.MathSciNetMATHCrossRefGoogle Scholar
  7. [D2]
    C. M. Dafermos, Contemporary Issues in the Dynamic Behavior of Continuous Media, LCDS Lecture Notes # 85–1, 1985.Google Scholar
  8. [DH1]
    C. M. Dafermos and L. Hsiao, Adiabatic shearing of incompressible fluids with temperature dependent viscosity, Quart. Appl. Math. 41 (1983), 45–58.MathSciNetADSMATHGoogle Scholar
  9. [DH2]
    C. M. Dafermos and L. Hsiao, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. Appl. Math. 44 (1986), 463–474.MathSciNetMATHGoogle Scholar
  10. [L]
    T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Amer. Math. Soc. Mem. 328 (1985).Google Scholar
  11. [N]
    A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Comp. Methods in Appl. Mech. and Engineering 67 (1988), 69–85.ADSMATHCrossRefGoogle Scholar
  12. [PW]
    M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1967.Google Scholar
  13. [SC]
    T. G. shawki and R. J. Clifton, Shear band formation in thermal viscoplastic materials, Mech. Mater. 8 (1989), 13–43.CrossRefGoogle Scholar
  14. [T1]
    A. E. Tzavaras, Shearing of materials exhibiting thermal softening or temperature dependent viscosity, Quart. Appl. Math. 44 (1986), 1–12.MathSciNetMATHGoogle Scholar
  15. [T2]
    A. E. Tzavaras, Plastic shearing of materials exhibiting strain hardening or strain softening, Arch. Rational Mech. Analysis 94 (1986), 39–58.MathSciNetADSMATHCrossRefGoogle Scholar
  16. [T3]
    A. E. Tzavaras, Effect of thermal softening in shearing of strain-rate dependent materials, Arch. Rational Mech. Analysis 99 (1987), 349–374.MathSciNetADSMATHCrossRefGoogle Scholar
  17. [T4]
    A. E. Tzavaras, Strain softening in viscoelasticity of the rate type, J. Integral Equations Appl. (to appear).Google Scholar
  18. [WW]
    T. W. Wright and J. W. Walter, On stress collapse in adiabatic shear bands, J. Mech. Phys. Solids 35 (1988), 701–720.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Athanasios E. Tzavaras
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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